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Further results on common zeros of the solutions of two differential equations

Abstract

Purpose

Two problems are discussed in this paper. In the first problem, we consider one homogeneous and one non-homogeneous differential equations and study when the solutions of these differential equations can have (nearly) the same zeros. In the second problem, we consider two linear second-order differential equations and investigate when the solutions of these differential equations can take the value 0 and a non-zero value at (nearly) the same points.

Method

We apply the Nevanlinna theory and properties of entire solutions of linear differential equations.

Conclusion

In the first problem, the results determine all pairs of such equations having solutions with the same zeros or nearly the same zeros. Regarding the second problem, the results also show all pairs of such equations having solutions taking the value 0 and a non-zero value at (nearly) the same points.

1 Introduction

There has been much research [18] on zeros of solutions of linear differential equations with entire coefficients. The principal paper [9] that was published in 1982 by Bank and Laine has stimulated many studies on this kind of problems. The reader is referred to [1012] for background on some applications of the Nevanlinna theory. We use the standard notation of the Nevanlinna theory from [13].

In 1955, Wittich [12] proved the following theorem.

Theorem 1.1 If f is a non-trivial solution of w +Aw=0, i.e., f0 and A0 is entire, then we have:

(i) T(r,A)=S(r,f).

(ii) If f has finite order, then A is a polynomial.

(iii) If a is a non-zero complex number, then f takes the value a infinitely often, and in fact, outside a set of finite measure,

N ( r , 1 f a ) T(r,f).

The following facts follow from the asymptotic representation for solutions of the equation

w +Pw=0.
(1)

Theorem 1.2 [9, 11]

Let P be a polynomial of degree n, and let w be a non-trivial solution of the equation (1). Then, w has order of growth equal to n + 2 2 . Moreover, if w is a solution of (1) which has infinitely many zeros, then we have

lim inf r N ( r , 1 w ) r ( n + 2 ) / 2 >0.
(2)

Our previous paper [14] studied homogeneous linear differential equations having solutions with nearly the same zeros and proved several results, including the following.

Theorem 1.3 [14]

Let P0 be a polynomial of degree n. Let w0 be a solution of (1). Assume that w has infinitely many zeros. Suppose that we have a solution v0 of the differential equation

v +Av=0
(3)

such that A is an entire function and N(r) counts zeros of v which are not zeros of w and zeros of w which are not zeros of v. Assume that

N(r)+T(r,A)=o ( r ( n + 2 ) / 2 ) .

Then v w is a constant and A=P.

The paper [14] includes further results for homogeneous linear differential equations, and the corresponding problem where P is a transcendental entire function of finite order is studied in [15].

Recently, the same problem, but with non-homogeneous first-order differential equations, has been studied in [16], including the following result.

Theorem 1.4 [16]

Assume that v =Av+B and w =Cw+D, where A, B, C and D are entire functions of order less than 1 and v, w are transcendental functions. Assume that v=Lw, where L has finitely many zeros and poles, and

T(r,A)+T(r,B)=S(r,v),T(r,C)+T(r,D)=S(r,w).
(4)

Then the following conclusions hold.

(I) If L is a rational function, then AC, L is a constant and B=LD.

(II) If L is a transcendental function, then one of the following cases holds: If, in addition, L has finite order in case (ii), then A, B, C, D are polynomials and so is A 1 .

(i) BD0 and v, w have no zeros.

(ii) A=C and B/A, D/C are non-zero constants, and

v= c 1 + c 2 e A 1 ,w= c 3 + c 4 e A 1 ,

where c j C, A 1 =A and L=(constant) e A 1 .

In this paper, our first result (Theorem 2.1) looks at the same problem but with one homogeneous and one non-homogeneous differential equations. In particular, we consider the first equation to be homogeneous of the second-order with a polynomial coefficient and the second equation to be non-homogeneous of the first-order with entire coefficients.

A further result (Theorem 2.2) studies the case where the solutions of two second-order homogeneous differential equations can take the value 0 and a non-zero value at (nearly) the same points.

2 Our results

Our first result is the following theorem.

Theorem 2.1 Suppose that P0 is a polynomial of degree n, and w solves (1), and w0 has infinitely many zeros. Suppose that v0 solves

v =Av+B,
(5)

where A, B are entire and AB0, and

T(r,A)+T(r,B)=o ( r ( n + 2 ) / 2 ) .
(6)

Suppose that L= w v has finitely many zeros and poles (i.e., w and v have the same zeros with finitely many exceptions).

Then A is a polynomial and there exists a polynomial Q such that

w= c 1 e Q e 2 Q dz and v= c 2 e A 1 e 2 Q dz,

and

P= ( Q + Q 2 ) andB= c 2 e A 1 2 Q ,

where A 1 =A and c 1 , c 2 are constants.

Example 2.1 Take Q to be a polynomial. Let

w= e Q 0 z e 2 Q dz.

Then

w = Q w+ e Q ,

and

w = Q w+ Q w Q e Q = Q w+ Q ( Q w + e Q ) Q e Q = ( Q + Q 2 ) w.

So, we have P=( Q + Q 2 ).

Now, let A 1 be another polynomial, and let

v= e A 1 0 z e 2 Q dz.

Note that v has the same zeros as w. Now, we have

v =Av+ e A 1 2 Q ,

where A= A 1 .

We choose A 1 so that

deg( A 1 2Q)< deg ( P ) + 2 2 .

For example, let A 1 =2Q.

We now state our second result.

Theorem 2.2 Suppose that P0 is a polynomial of degree n, and A is an entire function, and suppose that w solves (1) and v solves (3), and vw0. Let v1 and w have, with finitely many exceptions, the same zeros and the same multiplicities. Then one of the following holds.

(A) w has finitely many zeros and v is a polynomial and A=0.

(B) w has infinitely many zeros and P, A are non-zero constants and v 1 w is non-constant and

w = λ 1 e σ z + λ 2 e σ z , v = λ 3 e 2 σ z ,
(7)

where σ, λ 1 , λ 2 , λ 3 are non-zero constants.

Example 2.2 If w= e z e z and v= e 2 z , then

v1= e 2 z 1= e z ( e z e z ) = e z w.

Hence, v1 has the same zeros as w. Here P=1 and A=4.

Example 2.3 We give an example to show that the zeros of v1 and w must necessarily have the same multiplicities. To show this, let

w=sin z 2 ,v=cosz.

Then w=0 z 2 =kπ, where kZ.

Also v=1z=k2π, where kZ.

Therefore, w and v1 have the same zeros but the zeros are simple for w, double for v1. Here, P= 1 4 and A=1.

3 Proof of Theorem 2.1

Proof We have

w=Lv.
(8)

So,

w = L v+2 L v +L v .
(9)

but

v = A v+A v + B = A v+A(Av+B)+ B =v ( A 2 + A ) +AB+ B .
(10)

We also have, using (1), (5), (8), (9) and (10),

0 = w + P w = L v + 2 L v + L v + P L v = L v + 2 L ( A v + B ) + L [ v ( A 2 + A ) + A B + B ] + P L v = [ L + 2 L A + L ( A 2 + A ) + P L ] v + 2 L B + L ( A B + B ) .
(11)

Let

M= L L .
(12)

Then

L L = M + M 2 .
(13)

We divide (11) by L, and by using (12) and (13), we get

0 = [ L L + 2 L L A + A 2 + A + P ] v + 2 L L B + A B + B = [ M + M 2 + 2 M A + A 2 + A + P ] v + 2 M B + A B + B .
(14)

The next step is to estimate the growth of M.

We know that ρ(w)= n + 2 2 from [9]. Therefore,

m ( r , w w ) =O(logr)=o ( r ( n + 2 ) / 2 ) .
(15)

Claim 3.1 We claim that T(r,M)=o( r ( n + 2 ) / 2 ).

To show this, we know that N(r,m)=O(logr) since M has finitely many poles.

Write (5) as

( v e A 1 ) = e A 1 B,

where A 1 =A.

Then, there exists a constant c such that

v= e A 1 ( c + 0 z e A 1 ( t ) B ( t ) d t ) .

Also, using (6), we can write

logM(r,A)3T(2r,A)=o ( r ( n + 2 ) / 2 ) ,M(r,A)exp ( o ( r ( n + 2 ) / 2 ) ) .

Also,

A 1 (z)= A 1 (0)+ 0 z A(t)dt.

So,

M(r, A 1 ) | A 1 ( 0 ) | +rM(r,A)O(1)+rexp ( o ( r ( n + 2 ) / 2 ) ) exp ( o ( r ( n + 2 ) / 2 ) ) .

Therefore, we get

M(r,v)expexp ( o ( r ( n + 2 ) / 2 ) ) ,T(r,v)logM(r,v)exp ( o ( r ( n + 2 ) / 2 ) ) .

We use Lemma 2.3 in [[13], p.38] with R=2r to get

m ( r , v v ) =O(logr)+O ( log + T ( 2 r , v ) ) o ( r ( n + 2 ) / 2 ) .

Now, we have M= w w v v . So

m(r,M)=o ( r ( n + 2 ) / 2 ) .

We also have N(r,M)=O(logr).

Hence,

T(r,M)=o ( r ( n + 2 ) / 2 ) +O(logr)=o ( r ( n + 2 ) / 2 ) .

This completes the proof of Claim 3.1.

Using Claim 3.1 and (14), we get

T ( r , M + M 2 + 2 M A + A 2 + A + P ) =o ( r ( n + 2 ) / 2 )

and

T ( r , 2 M B + A B + B ) =o ( r ( n + 2 ) / 2 ) .

Also, by Theorem 1.2, we get

T(r,v)N ( r , 1 v ) O(1)(constant) r ( n + 2 ) / 2 .

Therefore, we must have

M 1 = M + M 2 +2MA+ A 2 + A +P0
(16)

and

M 2 =2MB+AB+ B 0
(17)

because otherwise we can write v= M 2 / M 1 to get a contradiction.

We now divide (17) by B to get

2M+A+ B B 0.

So, B /B has finitely many poles, and so B has finitely many zeros. Then we can write B in the form

B= P 1 e P 2 ,

where P 1 , P 2 are polynomials.

But then, we can write

B /B= R 1 ,

where R 1 is rational.

Then we also can write

M= A 2 + R 2 ,
(18)

where R 2 is rational and R 2 = 1 2 R 1 .

Substitute (18) in (16), we obtain

0 ( A 2 + R 2 ) + ( A 2 4 A R 2 + R 2 2 ) + ( A 2 + 2 A R 2 ) + A 2 + A + P ( A 2 + R 2 ) + ( A 2 4 + A R 2 + R 2 2 ) + P ( A 2 + R 2 ) + ( A 2 + R 2 ) 2 + P .
(19)

Now, let

N= A 2 + R 2 = A 2 B 2 B .

Also, let

H= e A 1 2 B 1 2 .
(20)

So, we get

H H = A 2 B 2 B =N.
(21)

Substituting (21) in (19), we obtain

0= N + N 2 +P= H H +P, H +PH=0.

Thus, ρ(H)<, H is entire, and B has no zeros.

Then,

m ( r , H H ) =O(logr)

Therefore, A is a polynomial.

Since B has no zeros, from (20) we can write

H= e Q ,
(22)

where Q is a polynomial.

Since w and H solve the same equation and are linearly independent (because w has zeros but H does not), we can write

( w H ) = (constant) H 2 .

Therefore,

w= c 1 e Q e 2 Q dz,
(23)

where c 1 is a constant and Q is a polynomial.

Now, we have

L L =M= A 2 + R 2 =NA= H H A.

So, we can write

L=(constant)H e A 1 .

Therefore, we have

v= w L = (constant) H H 2 d z H e A 1 .

Hence, using (22), we obtain

v= c 2 e A 1 e 2 Q dz,
(24)

where c 2 is a constant and A 1 =A.

Now, from (23) and (24), we notice that w and v have the same zeros.

Also, differentiating (24), using (22), we have

v = c 2 A e A 1 H 2 dz+ c 2 e A 1 H 2 =Av+ c 2 e A 1 H 2 =Av+ c 2 e A 1 2 Q .

Comparing this with (5), we get

B= c 2 e A 1 2 Q .

Moreover, H= e Q solves H +PH=0, and so

P= H H = Q + Q 2 .

This completes the proof of Theorem 2.1.  □

4 A lemma needed to prove Theorem 2.2

In order to prove Theorem 2.2, we must state and prove the following lemma. We include a proof for completeness.

Lemma 4.1 Let P 1 ,, P n C be distinct, and let A 1 ,, A n be rational functions such that

1 A 1 (z) e P 1 z ++ A n (z) e P n z .
(25)

Then there exists k{1,,n} such that P k =0 and A k =1, and A j =0 for jk.

Proof The proof is by induction. It is obvious that the lemma is true when n=1.

Assume that the lemma is true for mn1. Differentiating (25), we get

0 B 1 (z) e P 1 z ++ B n (z) e P n z , B j = A j + P j A j .

Now, we have two cases to consider.

Case (1): Suppose there exists k such that B k 0. Without loss of generality, let k=1, then we can write

0=1+ B 2 B 1 e ( P 2 P 1 ) z ++ B n B 1 e ( P n P 1 ) z .

Since we assumed the lemma is true for mn1, there exists j{2,,n} such that P j P 1 =0. But this contradicts our assumption that P 1 ,, P n are distinct.

Case (2): Suppose that B j =0 for each j, i.e.,

A j + P j A j 0.

If P j 0, then A j 0 because otherwise we have

A j A j + P j 0, A j (z) e P j z =cC{0}.

But this contradicts the fact that P j 0.

So, we have A j 0 for P j 0. Thus, (25) becomes (for some k)

1= A k (z) e 0 z = A k (z),

and P k =0 and A k =1. □

5 Proof of Theorem 2.2

We first note that, outside a set of finite measure, by Theorem 1.1,

T(r,v)N ( r , 1 v 1 ) N ( r , 1 w ) +O(logr)O ( r ( n + 2 ) / 2 ) +O(logr).
(26)

In particular, if w has finitely many zeros, then v is a polynomial, which gives A=0. This completes the proof of part (A) in the conclusion.

Assume henceforth that w has infinitely many zeros. Then (26) implies that ρ(v)(n+2)/2, and so A is a polynomial of degree at most n by the Wiman-Valiron theory [17]. Also, A0 since v1 has infinitely many zeros.

Now, two cases have to be considered.

Case (I). Assume that P is a non-zero constant; then n=0 and A is constant. Therefore, we can write

w= c 1 e α z + c 2 e α z ,v= d 1 e β z + d 2 e β z ,
(27)

where α,βC{0}, c j , d j C and c j 0 (j=1,2).

Since w and v1 have the same zeros with finitely many exceptions, we can write

v1= R 1 e P 1 w,
(28)

where R 1 is a rational function and P 1 is a polynomial. We know that deg( P 1 )1 because ρ(w),ρ(v)1. We can now write

d 1 e β z + d 2 e β z 1= R 1 e γ z ( c 1 e α z + c 2 e α z ) ,

where γC, and so

1= d 1 e β z + d 2 e β z c 1 R 1 e ( γ + α ) z c 2 R 1 e ( γ α ) z .

Now, by using Lemma 4.1, R 1 is constant and so we can write (28) as

v 1 w = e γ z + δ ,
(29)

where δ is constant.

Therefore,

d 1 e β z + d 2 e β z 1= ( e γ z + δ ) ( c 1 e α z + c 2 e α z ) = c 1 e ( α + γ ) z + δ + c 2 e ( α + γ ) z + δ .
(30)

Now, by using Lemma 4.1, we get

α+γ=β,β or 0,α+γ=β,β or 0,

and α+γ, α+γ, β, −β, 0 cannot all be different.

We must now try six cases:

I(a): If α+γ=β and α+γ=β, then γ=0 and α=β. But this contradicts (30). Thus, this case cannot happen.

I(b): If α+γ=β and α+γ=0, then β=2γ and α=γ. Substituting these in (30) gives

d 1 e 2 γ z + d 2 e 2 γ z 1= h 1 e 2 γ z + h 2 ,

where h 1 , h 2 are constants, which yields d 2 =0. Putting this in (27) gives (7) with σ=γ.

There are four more cases:

I(c): α+γ=β and α+γ=β.

I(d): α+γ=β and α+γ=0.

I(e): α+γ=0 and α+γ=β.

I(f): α+γ=0 and α+γ=β.

It is easy to check that case I(c) is impossible and cases I(d), I(e), I(f) all lead to (7) with σ=γ.

From these cases, we find that γ0, and so v 1 w = e γ z + δ is non-constant. Also, we have (7) and case (B) of the conclusion.

Case (II). Suppose that P is non-constant. We will show that this leads to a contradiction. Let

v 1 w =M=L e Q ,
(31)

where L is a rational function and Q is an entire function.

From (26), we have ρ(v)<, and so Q is a polynomial.

Also, from (31), we have

So,

2 M w + ( M + A M P M ) w=A.
(32)

Now, we have two cases to consider.

Case (i): If M is constant, then either A=P and A=0, so that P=0, which is a contradiction, or

w= A A M P M

is a rational function, which is a contradiction since w has infinitely many zeros.

Case (ii): If M is non-constant, then M 0. Therefore,

w + ( M 2 M + ( A P ) M 2 M ) w= A 2 M ,
(33)

where M 2 M + ( A P ) M 2 M is rational because M M = L L + Q is rational and M M is rational, and so M M = M M / M M is rational.

Also,

A 2 M = A 2 ( L + Q L ) e Q .

Then we can write (33) as

w =Rw+S e Q ,
(34)

where

R= P M A M M 2 M ,S= A 2 ( L + Q L )

are rational functions and Q is a polynomial.

Let U=S e Q , then we can write (34) as

w =Rw+U.
(35)

Now, we have two cases to consider:

Case ii(a): If R0 in (35), then (34) gives

w =S e Q ,w= w P = ( S Q S ) P e Q ,

which is a contradiction since w has infinitely many zeros.

Case ii(b): Assume that R0 in (35); then (35) gives

w =R w + R w+ U .

Now, (1) and (35) give

Pw=R(Rw+U)+ R w+ U ,

and so

( R + R 2 + P ) w+RU+ U =0.

Therefore,

R + R 2 +P0,
(36)

because if not, w has finitely many zeros, a contradiction. Also,

RU+ U 0.
(37)

Put

G= 1 U =T e Q ,
(38)

where T=1/S is a rational function.

Then,

R= U U = G G .

From (36), we get

0= R + R 2 +P= G G +P, G +PG=0.

So, G solves (1), and since P0 and is a polynomial of degree n, we see that G is a transcendental entire function with finitely many zeros and has order (n+2)/2.

Since w and G solve the same equation but w has infinitely many zeros and G has finitely many zeros, w and G are linearly independent, and we can write

w Gw G =c,

where c is a non-zero constant. So,

( w G ) = c G 2 .

By integrating, we get

w=G z c G 2 dζ.
(39)

Also, using (31), (38) and (39),

v=1+Mw=1+L e Q T e Q z c G 2 dζ=1+H G 2 z c G 2 dζ,
(40)

where H=L/T is a rational function.

Now, we can assume that c=1 because if c1, we can multiply w by 1/c.

We differentiate (40) to get

v = H G 2 z 1 G 2 dζ+2HG G z 1 G 2 dζ+H=H+K(v1),
(41)

where

K= H H +2 G G
(42)

is a rational function.

So, from (3) and (41), we get

Av= v = H + K (v1)+K [ H + K ( v 1 ) ] ,

and so

0=v ( K + K 2 + A ) + ( H K + K H K 2 ) =v U 1 + U 2 ,

where U 1 = K + K 2 +A and U 2 = H K +KH K 2 .

Since v is transcendental and U 1 , U 2 are rational functions, we must have

U 1 = K + K 2 +A=0
(43)

and

U 2 = H K +KH K 2 =0.
(44)

Claim 5.1 We claim that HK.

To show this, let HK.

From (44), we have

H K H K +K=0.

From (42), we get

H K H K =K= H H 2 G G .

We integrate to get

HK= a H G 2 , G 2 = a H ( H K ) ,

where a is a constant. But this contradicts the fact that H and K are rational functions and G is a transcendental function. This completes the proof of Claim 5.1.

Once we have Claim 5.1, (41) gives

v =H+H(v1)=Hv,

and so

H= v v .
(45)

By (45), v has finitely many zeros, so we can write

v= P 1 e Q 1 ,

where P 1 , Q 1 are polynomials, P 1 0, and Q 1 is non-constant because v is transcendental.

Therefore,

w= v 1 L e Q = P 1 e Q 1 1 L e Q .

Now, we can write this as

w= R 1 e S 1 + R 2 e S 2 ,
(46)

where R 1 = P 1 /L0, R 2 =1/L0 are rational functions and S 1 = Q 1 Q, S 2 =Q are polynomials.

Here, R 1 e S 1 and R 2 e S 2 are linearly independent because Q 1 is non-constant. Now, we get

0= w +Pw= J 1 e S 1 + J 2 e S 2 = J 1 e Q 1 Q + J 2 e Q ,

where J 1 , J 2 are rational and satisfy

J k = R k +2 R k S k + R k ( S k + S k 2 + P ) .

Therefore, J 1 = J 2 =0 because otherwise e Q 1 = J 2 / J 1 or e Q 1 = J 1 / J 2 . Thus, R 1 e S 1 , R 2 e S 2 both solve y +Py=0 and have finitely many zeros, and they are linearly independent.

Hence, P is constant by [9], which contradicts our assumption in Case (II) that P is non-constant. □

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Acknowledgements

The author would like to thank his supervisor Prof. Jim Langley for his support and guidance. Also, he would like to thank King Abdulaziz University for financial support for his PhD study.

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Asiri, A. Further results on common zeros of the solutions of two differential equations. J Inequal Appl 2012, 222 (2012). https://doi.org/10.1186/1029-242X-2012-222

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Keywords

  • Nevanlinna theory
  • differential equations